Axiom of reducibility

The axiom of reducibility was introduced by Bertrand Russell as part of his ramified theory of types, an attempt to ground mathematics in first-order logic.

The axiom of reducibility is introduced in number (chapter) *12 of Principia Mathematica (1910) as a pair of primitive propositions *12.1 and *12.11 [Page 167 "Principia Mathematica to *56", Cambridge Mathematical Library, 1997] . In 1925 Frank Plumpton Ramsey argued that it is not needed [The Foundations of Mathematics (1925), pages 1..61 of "The Foundations of Mathematics", F. P. Ramsey, Littlefield Adams & Co, Paterson New Jersey, 1960] . However in the second edition of Principia Mathematica (1927, page xiv) and in Ramsey's 1926 paper [Mathematical Logic, pages 62..61, op. cit. ] it is stated that certain theorems about real numbers could not be proved using Ramsey's approach. Most later mathematical formalisms (Hilbert's Formalism (mathematics) or Brower's Intuitionism for example) do not use it. It is an ox-bow lake in the river of modern formal logic and meta-mathematics.

The axiom of reducibility states that any truth function can be expressed by a formally equivalent "predicative" truth function. Whitehead and Russell defined predicative functions as functions whose order is one more than the highest order function occurring in the expression of the function(page 58 of "Principia Mathematica"). They are introduced by Whitehead and Russell to avoid certain vicious circle paradoxes that are not handled by the theory of types (Introduction, Chapter II, Sections V through VII). Ramsey showed that you can reformulate the definition of "predicative" by using the definitions in Wittgenstein's Tractatus Logico-Philosophicus. As a result, all functions of a given order are "predicative", irrespective of how they are expressed. He goes on to show that his formulation still avoids the paradoxes. However, the "Tractatus" theory did not appear strong enough to prove some mathematical results.

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